Question

Body Fat. In the paper "Total Body Composition by DualPhoton ( ${}^{153}$Gd) Absorptiometry" (American Journal of Clinical Nutrition, Vol. 40, pp. 834-839), R. Mazess et al. studied methods for quantifying body composition. Eighteen randomly selected adults were measured for percentage of body fat, using dual-photon absorptiometry. Each adult's age and percentage of body fat are shown on the WeissStats site.

Short answer

(a) For the provided data, the regression t-test is appropriate.

(b) The data support the conclusion that the predictor variable "adult ages" is beneficial for predicting "body fat" at the $5\%$level.

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that, Body Fat. In the paper "Total Body Composition by DualPhoton ( ${}^{153}$ Gd) Absorptiometry" (American Journal of Clinical Nutrition, Vol. 40, pp. 834-839), R. Mazess et al. studied methods for quantifying body composition. Eighteen randomly selected adults were measured for percentage of body fat, using dual-photon absorptiometry. Each adult's age and percentage of body fat are shown on the WeissStats site. We need to decide that whether we can reasonably apply the regression t-lest. If so, then also do part (b).

Part (a) Step 2: Explanation

AGE	%FAT
$23$	$9.5$
$23$	27.9
$27$	$17.8$
$39$	$31.4$
$41$	$25.9$
$45$	$27.4$
$49$	$25.2$
$50$	$31.1$
$53$	$34.7$
$53$	$42$
$54$	$29.1$
$56$	$32.5$
$50$	$30.3$
$58$	$33$
$58$	$33.8$
$60$	41.1
$61$	$34.5$

MINITAB is used to create the residual plot.

Procedure for MINITAB:

Step 1: Select Stat > Regression > Regression from the drop-down menu.

Step 2: In the Response box, type percent Fat.

Step 3: In Predictors, fill in the Age columns.

Step 4: In Graphs, under Residuals vs the variables, enter the columns Age.

Step 5: Click the OK button

OUTPUT FROM MINITAB:

Part(a) Step 3: Construct the residual plot

Using the MINITAB technique, create a normal probability plot of residuals:

Step 1: Select Stat > Regression > Regression from the drop-down menu.

Step 2: In the Response box, type percent Fat.

Step 3: In Predictors, fill in the Age columns.

Step 4: Select Normal probability plot of residuals from the Graphs menu.

Step 5: Click the OK button.

OUTPUT FROM MINITAB:

The following is the assumption for regression inferences: Line of population regression:

For each value $X$of the predicator variable, the conditional mean of the response variable $\left(Y\right)$is ${\beta }_0+{\beta }_{1}X$.

Standard deviations are equal:

The response variable's $\left(Y\right)$standard deviation is the same as the explanatory variable's$\left(X\right)$standard deviation. The standard deviation is represented by the symbol $\sigma$.

Typical populations include:

The response variable follows a normal distribution.

Observations made independently:

The response variable observations are unrelated to one another.

Examine whether the graph shows a violation of one or more of the regression inference assumptions.

• It is obvious from the residual plot that the residuals fall into the horizontal band.
• It is obvious from the normal probability plot of residuals that

Part (b) Step 1: Given information

Given in the question that , In the paper "Total Body Composition by DualPhoton ( $\left.{}^{153}\mathrm{Gd}\right)$ Absorptiometry" (American Journal of Clinical Nutrition, Vol. 40, pp. 834-839), R. Mazess et al. studied methods for quantifying body composition. Eighteen randomly selected adults were measured for percentage of body fat, using dual-photon absorptiometry. Each adult's age and percentage of body fat are shown on the Weiss Stats site. We need to decide, at the$5\%$ signiflcance level, whether the data provide sufflcient evidence to conclude that the predictor variable is useful for predicting the response variable.

Part (b) Step 2: Explanation

The following are the suitable hypotheses:

$H_0:{\beta }_1=0$

Null hypothesis:

In other words, the predictor variable "age" is ineffective in predicting "$\%fat$."

Alternative hypothesis:

$H_{\alpha }:{\beta }_1\ne 0$

That example, the predictor variable "age" can be used to forecast "$\%$fat"

Rule of Rejection: If the p-value $\le \alpha (=0.05)$, reject the null hypothesis $\left(H_0\right)$.

MINITAB can be used to find the test statistic and p-value.

Procedure for MINITAB:

Step 1: Select Stat > Regression > Regression from the drop-down menu.

Step 2: In the Response box, type percent Fat.

Step 3: In Predictors, fill in the Age columns.

Step 4: Click the OK button.

MINITAB output:

Regression Analysis:$\%$FAT versus AGE

Model Summary

The test statistic value is $5.19$, and the $p$-value is $0.000$, according to the MINITAB report.

Conclusion: Use the $\alpha =0.05$significance threshold.

The p-value is lower than the level of significance in this case.

Specifically, $p$-value $(=0.000)<\alpha (=0.05)$.

According to the rejection criterion, there is sufficient evidence to reject the null hypothesis ($\left.H_0\right)$at $\alpha =0.05$.